A143839 Ulam's spiral (SSE spoke).

1, 24, 79, 166, 285, 436, 619, 834, 1081, 1360, 1671, 2014, 2389, 2796, 3235, 3706, 4209, 4744, 5311, 5910, 6541, 7204, 7899, 8626, 9385, 10176, 10999, 11854, 12741, 13660, 14611, 15594, 16609, 17656, 18735, 19846, 20989, 22164, 23371, 24610

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 02 2008

Keywords

• nonn
• easy

Comments

Also sequence found by reading the line from 1, in the direction 1, 24, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A004767, A016813.

Programs

GAP

List([1..40], n-> ((32*n -25)^2 +15)/64); # G. C. Greubel, Nov 09 2019

Magma

[16*n^2-25*n+10: n in [1..40]]; // Vincenzo Librandi, Nov 08 2014

Maple

seq( ((32*n -25)^2 +15)/64, n=1..40); # G. C. Greubel, Nov 09 2019

Mathematica

f[n_] := 16n^2 -25n +10; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
LinearRecurrence[{3,-3,1},{1,24,79},40] (* Harvey P. Dale, May 26 2012 *)
CoefficientList[Series[(1+21*x+10*x^2)/(1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 08 2014 *)

PARI

Vec(x*(1+21*x+10*x^2)/(1-x)^3 + O(x^40)) \\ Colin Barker, Nov 08 2014

Sage

[((32*n -25)^2 +15)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

Formula

a(n) = 16*n^2 - 25*n + 10. - R. J. Mathar, Sep 04 2008

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1) = 1, a(2) = 24, a(3) = 79. - Harvey P. Dale, May 26 2012

G.f.: x*(1 + 21*x + 10*x^2)/(1-x)^3. - Harvey P. Dale, May 26 2012

E.g.f.: exp(x)*(10 - 9*x + 16*x^2) - 9. - Stefano Spezia, Oct 07 2019